I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. MATERIALS AND METHODSIII. RESULTS AND DISCUSSI...IV. CONCLUSIONSSUPPLEMENTARY MATERIALREFERENCESOxygen delivery plays a vital role in a vast number of cellular processes and is transported via hemoglobin in red blood cells (RBCs) from the lungs into the microcirculation. There, along with nutrients, oxygen diffuses into the tissues for use in metabolic and cellular processes.11. S. S. Shevkoplyas, S. C. Gifford, T. Yoshida, and M. W. Bitensky, “Prototype of an in vitro model of the microcirculation,” Microvasc. Res. 65(2), 132–136 (2003). https://doi.org/10.1016/S0026-2862(02)00034-1 The microcirculation is a complex network of bifurcating and diverging vessels, many of which have diameters that are on the size scale of the RBC diameter. Several severe pathological states, such as Alzheimer's disease22. J. de la Torre, “Evidence that Alzheimer’s disease is a microvascular disorder: The role of constitutive nitric oxide,” Brain Res. Rev. 34(3), 119–136 (2000). https://doi.org/10.1016/S0165-0173(00)00043-6,33. E. Gutiérrez-Jiménez, H. Angleys, P. M. Rasmussen et al., “Disturbances in the control of capillary flow in an aged APPswe/PS1ΔE9 model of Alzheimer’s disease,” Neurobiol. Aging 62, 82–94 (2018). https://doi.org/10.1016/j.neurobiolaging.2017.10.006 and coronary microvascular disease,44. P. G. Camici and F. Crea, “Coronary microvascular dysfunction,” N. Engl. J. Med. 356(8), 830–840 (2007). https://doi.org/10.1056/NEJMra061889 have been associated with disturbances in the microvasculature network topology. The abnormal vascular morphology in tumors has also been proposed as a cause of the atypical tissue oxygenation, which is often observed.55. M. O. Bernabeu, J. Köry, J. A. Grogan et al., “Abnormal morphology biases hematocrit distribution in tumor vasculature and contributes to heterogeneity in tissue oxygenation,” Proc. Natl. Acad. Sci. U.S.A. 117(45), 27811–27819 (2020). https://doi.org/10.1073/pnas.2007770117 Additionally, there is a multitude of other pathologies such as sickle cell anemia,66. K. Dufu, M. Patel, D. Oksenberg, and P. Cabrales, “GBT440 improves red blood cell deformability and reduces viscosity of sickle cell blood under deoxygenated conditions,” Clin. Hemorheol. Microcirc. 70(1), 95–105 (2018). https://doi.org/10.3233/CH-170340 malaria,77. M. Depond, B. Henry, P. Buffet, and P. A. Ndour, “Methods to investigate the deformability of RBC during malaria,” Front. Phys. 10, 01613 (2020). https://doi.org/10.3389/fphys.2019.01613 and diabetes,88. R. Agrawal, T. Smart, J. Nobre-Cardoso et al., “Assessment of red blood cell deformability in type 2 diabetes mellitus and diabetic retinopathy by dual optical tweezers stretching technique,” Sci. Rep. 6(1), 15873 (2016). https://doi.org/10.1038/srep15873 which result in severely impaired RBC deformability, reducing the ability of RBCs to perfuse through the smaller vessels of the microvasculature.In larger vessels, blood perfusion can be analyzed as bulk fluid flow with the blood apparent viscosity modeled as a Newtonian or non-Newtonian fluid depending on the vessel geometry and flow conditions, the complexity of the analysis, and rheological constitutive model chosen (i.e., Newtonian, power-law, Casson, Quemada models, etc.). By contrast, perfusion within the microvasculature occurs as a two-phase suspension of highly deformable cells within the Newtonian plasma phase. The blood flow behavior in the microcirculation can lead to heterogeneous hematocrit distribution through the network and even RBC-free vessels. This heterogeneous cell distribution has been attributed to RBC partitioning during perfusion through the multiple bifurcations of blood vessels found within the microcirculation.99. A. Mantegazza, F. Clavica, and D. Obrist, “In vitro investigations of red blood cell phase separation in a complex microchannel network,” Biomicrofluidics 14(1), 014101 (2020). https://doi.org/10.1063/1.5127840When RBCs perfuse through bifurcating vessels, the higher flow rate daughter channel often receives a higher erythrocyte flux in relation to the volumetric flow rate. This is called the bifurcation law, the Zweifach–Fung (ZF) effect,1010. K. Svanes and B. W. Zweifach, “Variations in small blood vessel hematocrits produced in hypothermic rats by micro-occlusion,” Microvasc. Res. 1(2), 210–220 (1968). https://doi.org/10.1016/0026-2862(68)90019-8 or plasma skimming.1111. A. R. Pries, T. W. Secomb, P. Gaehtgens, and J. F. Gross, “Blood flow in microvascular networks. Experiments and simulation,” Circ. Res. 67(4), 826–834 (1990). https://doi.org/10.1161/01.RES.67.4.826,1212. A. R. Pries, K. Ley, M. Claassen, and P. Gaehtgens, “Red cell distribution at microvascular bifurcations,” Microvasc. Res. 38(1), 81–101 (1989). https://doi.org/10.1016/0026-2862(89)90018-6 Initially described by Svanes and Zweifach,1010. K. Svanes and B. W. Zweifach, “Variations in small blood vessel hematocrits produced in hypothermic rats by micro-occlusion,” Microvasc. Res. 1(2), 210–220 (1968). https://doi.org/10.1016/0026-2862(68)90019-8 it was observed that by compressing blood vessels in a rat mesosecum, the vessel that had a reduction in the flow rate received disproportionately fewer RBCs than the higher flow rate vessel. Pries et al.1212. A. R. Pries, K. Ley, M. Claassen, and P. Gaehtgens, “Red cell distribution at microvascular bifurcations,” Microvasc. Res. 38(1), 81–101 (1989). https://doi.org/10.1016/0026-2862(89)90018-6 also conducted in vivo rat experiments and developed a sigmoidal fit between the daughter vessel flow rate ratio and RBC partitioning. Under specific conditions, such as a multi-file-type cell flow in the feeding vessel, the opposite partitioning can also occur, known as reverse partitioning, the higher flow rate channel receives a lower erythrocyte flux in relation to the volumetric flow rate.1313. P. Balogh and P. Bagchi, “Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks,” Phys. Fluids 30(5), 051902 (2018). https://doi.org/10.1063/1.5024783 As vessel diameter decreases, these effects are amplified and are a major contributor to the heterogeneous distribution of erythrocytes throughout the microvascular network where the ZF effect is most prominent.1414. T. W. Secomb, “Blood flow in the microcirculation,” Annu. Rev. Fluid Mech. 49(1), 443–461 (2017). https://doi.org/10.1146/annurev-fluid-010816-060302There is a vast amount of research modeling microvascular perfusion in silico, examining RBC perfusion and partitioning along anatomically accurate microvascular network models. Schmid-Schönbein et al.1515. G. W. Schmid-Schönbein, R. Skalak, S. Usami, and S. Chien, “Cell distribution in capillary networks,” Microvasc. Res. 19(1), 18–44 (1980). https://doi.org/10.1016/0026-2862(80)90082-5 quantified in vivo RBC partitioning data from a single capillary bifurcation and developed a cell distribution function of network partitioning, which is strongly non-linear in capillaries due to the particulate nature of blood in capillaries. Kiani et al.1616. M. F. Kiani, A. R. Pries, L. L. Hsu, I. H. Sarelius, and G. R. Cokelet, “Fluctuations in microvascular blood flow parameters caused by hemodynamic mechanisms,” Am. J. Phys. Heart Circ. Physiol. 266(5), H1822–H1828 (1994). https://doi.org/10.1152/ajpheart.1994.266.5.H1822 simulated temporal variations in local flow parameters, identifying a periodic variation of hematocrit in converging vessels. Carr et al.1717. R. T. Carr, J. B. Geddes, and F. Wu, “Oscillations in a simple microvascular network,” Ann. Biomed. Eng. 33(6), 764–771 (2005). https://doi.org/10.1007/s10439-005-2345-2 built upon Kiani's work and identified the simplest network topology under which hematocrit oscillations will occur, and later, Geddes et al.1818. J. B. Geddes, R. T. Carr, F. Wu, Y. Lao, and M. Maher, “Blood flow in microvascular networks: A study in nonlinear biology,” Chaos Interdiscip. J. Nonlinear Sci. 20(4), 045123 (2010). https://doi.org/10.1063/1.3530122 explored how topological complexity affects partitioning within the network. Balogh and Bagch13,1913. P. Balogh and P. Bagchi, “Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks,” Phys. Fluids 30(5), 051902 (2018). https://doi.org/10.1063/1.502478319. P. Balogh and P. Bagchi, “Direct numerical simulation of cellular-scale blood flow in 3D microvascular networks,” Biophys. J. 113(12), 2815–2826 (2017). https://doi.org/10.1016/j.bpj.2017.10.020 identified cell scale phenomena including hematocrit skewness and the positioning of cells along the inner wall of the vessel rather than being evenly distributed across the vessel in the feeder vessel as a major contributor to heterogeneous erythrocyte partitioning.In contrast to the vast amount of the literature available regarding in silico modeling of RBC partitioning within complex models of the multiple serial and parallel bifurcations found in the microvasculature, there are limited experimental studies utilizing networks in vitro, particularly, when the channel hydraulic diameter approaches that of an RBC. Prior in vitro partitioning experiments have infused RBCs,9,20–249. A. Mantegazza, F. Clavica, and D. Obrist, “In vitro investigations of red blood cell phase separation in a complex microchannel network,” Biomicrofluidics 14(1), 014101 (2020). https://doi.org/10.1063/1.512784020. S. Roman, A. Merlo, P. Duru, F. Risso, and S. Lorthois, “Going beyond 20 μm-sized channels for studying red blood cell phase separation in microfluidic bifurcations,” Biomicrofluidics 10(3), 034103 (2016). https://doi.org/10.1063/1.494895521. B. M. Fenton, R. T. Carr, and G. R. Cokelet, “Nonuniform red cell distribution in 20 to 100 μm bifurcations,” Microvasc. Res. 29(1), 103–126 (1985). https://doi.org/10.1016/0026-2862(85)90010-X22. F. Clavica, A. Homsy, L. Jeandupeux, and D. Obrist, “Red blood cell phase separation in symmetric and asymmetric microchannel networks: Effect of capillary dilation and inflow velocity,” Sci. Rep. 6(1), 36763 (2016). https://doi.org/10.1038/srep3676323. J. M. Sherwood, D. Holmes, E. Kaliviotis, and S. Balabani, “Spatial distributions of red blood cells significantly alter local haemodynamics,” PLoS One 9(6), e100473 (2014). https://doi.org/10.1371/journal.pone.010047324. A. Pskowski, P. Bagchi, and J. D. Zahn, “Investigation of red blood cell partitioning in an in vitro microvascular bifurcation,” Artif. Organs 45, 1083–1096 (2021). https://doi.org/10.1111/aor.13941 microparticles,2525. B. W. Roberts and W. L. Olbricht, “Flow-induced particulate separations,” AIChE J. 49(11), 2842–2849 (2003). https://doi.org/10.1002/aic.690491116,2626. B. W. Roberts and W. L. Olbricht, “The distribution of freely suspended particles at microfluidic bifurcations,” AIChE J. 52(1), 199–206 (2006). https://doi.org/10.1002/aic.10613 and lipid vesicles2727. V. Doyeux, T. Podgorski, S. Peponas, M. Ismail, and G. Coupier, “Spheres in the vicinity of a bifurcation: Elucidating the Zweifach–Fung effect,” J. Fluid Mech. 674, 359–388 (2011). https://doi.org/10.1017/S0022112010006567 through bifurcations, mainly in microfluidic networks, while varying feeder hematocrit, bifurcation geometry, and channel size. Sherwood et al.2323. J. M. Sherwood, D. Holmes, E. Kaliviotis, and S. Balabani, “Spatial distributions of red blood cells significantly alter local haemodynamics,” PLoS One 9(6), e100473 (2014). https://doi.org/10.1371/journal.pone.0100473 utilized 50 × 50 μm2 channels with sequential 90° bifurcations and observed that the bulk hematocrit was skewed toward the inner channel wall after perfusing through a bifurcation, drastically changing the local hematocrit distribution across the channel. Shen et al.2828. Z. Shen, G. Coupier, B. Kaoui, “Inversion of hematocrit partition at832 microfluidic bifurcations,” Microvasc. Res. 105, 40–46 (2016). https://doi.org/10.1016/j.mvr.2015.12.009 utilized a network consisting of a 20 × 8 μm2 channel with a single bifurcation followed by vessel convergence, demonstrating that RBC deformability and flow configuration affect partitioning within the bifurcation. Zhou et al.2929. Q. Zhou, J. Fidalgo, M. O. Bernabeu, M. S. N. Oliveira, and T. Krüger, “Emergent cell-free layer asymmetry and biased haematocrit partition in a biomimetic vascular network of successive biofurcations,” Soft. Matter 17, 3619–3633 (2021). https://doi.org/10.1039/D0SM01845G used both in silico and microfluidic experiments in a network with 30 × 30 μm2 channels and larger to investigate RBC partitioning, finding that heterogeneity increased at the upstream bifurcations with decreased flow rate. However, much of this work used simple network topology, namely, single bifurcation geometry.Experiments conducted in in vitro networks with dimensions comparable to RBCs include Shevakopylys et al.,11. S. S. Shevkoplyas, S. C. Gifford, T. Yoshida, and M. W. Bitensky, “Prototype of an in vitro model of the microcirculation,” Microvasc. Res. 65(2), 132–136 (2003). https://doi.org/10.1016/S0026-2862(02)00034-1 who developed an in vitro network model of the microcirculation and utilized it for a variety of studies including measuring the rheological properties of stored RBCs3030. J. M. Burns, X. Yang, O. Forouzan, J. M. Sosa, and S. S. Shevkoplyas, “Artificial microvascular network: A new tool for measuring rheologic properties of stored red blood cells,” Transfusion 52(5), 1010–1023 (2012). https://doi.org/10.1111/j.1537-2995.2011.03418.x and confirming the existence of spontaneous oscillations of blood flow determined by oscillatory erythrocyte velocity, within the capillaries.3131. O. Forouzan, X. Yang, J. M. Sosa, J. M. Burns, and S. S. Shevkoplyas, “Spontaneous oscillations of capillary blood flow in artificial microvascular networks,” Microvasc. Res. 84(12), 123–132 (2012). https://doi.org/10.1016/j.mvr.2012.06.006 Losserand et al.3939. S. Losserand, G. Coupier, and T. Podgorski, “Migration velocity of red blood cells in microchannels,” Microvasc. Res. 124, 30–36 (2019). https://doi.org/10.1016/j.mvr.2019.02.003 perfused RBCs through straight microchannels of varying widths, tracking RBC trajectory and analyzing lateral migration velocity. Stauber et al.3232. H. Stauber, D. Waisman, N. Korin, and J. Sznitman, “Red blood cell dynamics in biomimetic microfluidic networks of pulmonary alveolar capillaries,” Biomicrofluidics 11(1), 014103 (2017). https://doi.org/10.1063/1.4973930 developed an in vitro pulmonary capillary network model and observed that although at the microscale, RBC dynamics were heterogeneous and there was a linear relationship between the time averaged bulk flow rate and pressure drop as would be expected in a homogeneous fluid. However, none of the above works included analyses of RBC partitioning.Recently, Mantegazza et al.99. A. Mantegazza, F. Clavica, and D. Obrist, “In vitro investigations of red blood cell phase separation in a complex microchannel network,” Biomicrofluidics 14(1), 014101 (2020). https://doi.org/10.1063/1.5127840 perfused RBCs suspended at 10% hematocrit through an idealized honeycomb network consisting of 10 × 8 × 85 μm3 bifurcating and converging channels. It was observed that bifurcations closer to the inlet closely followed Pries’ separation law. Bifurcations further along the network deviated from the law to the point where reverse partitioning was occurring. This was then correlated with local hematocrit skewness within the respective feeder channel, providing further support to Balogh's earlier hypothesis that skewness is a main determinant of reverse partitioning.1313. P. Balogh and P. Bagchi, “Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks,” Phys. Fluids 30(5), 051902 (2018). https://doi.org/10.1063/1.5024783 While important, this study only varied driving pressure; further studies are needed to gain a more complete understanding of reverse partitioning as well as the magnitude of the effect of hematocrit skewness.Though the above studies have provided a basis for understanding blood perfusion within the microvasculature, there are still further experimental studies, which can be undertaken to understand more complex microvascular networks. While the effects of feeder hematocrit20,2420. S. Roman, A. Merlo, P. Duru, F. Risso, and S. Lorthois, “Going beyond 20 μm-sized channels for studying red blood cell phase separation in microfluidic bifurcations,” Biomicrofluidics 10(3), 034103 (2016). https://doi.org/10.1063/1.494895524. A. Pskowski, P. Bagchi, and J. D. Zahn, “Investigation of red blood cell partitioning in an in vitro microvascular bifurcation,” Artif. Organs 45, 1083–1096 (2021). https://doi.org/10.1111/aor.13941 and daughter channel flow rate ratio2424. A. Pskowski, P. Bagchi, and J. D. Zahn, “Investigation of red blood cell partitioning in an in vitro microvascular bifurcation,” Artif. Organs 45, 1083–1096 (2021). https://doi.org/10.1111/aor.13941 on erythrocyte partitioning have been extensively probed in single bifurcations, they have not been addressed in sequential bifurcations, which are a more realistic representation of the microvasculature. Furthermore, there is limited data on the effects of both RBC velocity and daughter channel length on erythrocyte partitioning within microchannels with diameters of the same size scale as RBCs.

In this study, idealized sequential bifurcating networks were utilized to investigate RBC partitioning. Human RBCs were perfused through 8 and 6 μm wide by 6.5 μm tall microchannels, similar to microvasculature vessel dimensions. The effects of feeder hematocrit, flow rate ratio, feeder channel length, and fluid velocity were all investigated. We demonstrate that feeder channel hematocrit skewness can significantly affect erythrocyte partitioning, causing extremes in reverse and regular partitioning and overcoming the previously observed effects of the daughter channel flow rate ratio on partitioning.

II. MATERIALS AND METHODS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. MATERIALS AND METHODS <<III. RESULTS AND DISCUSSI...IV. CONCLUSIONSSUPPLEMENTARY MATERIALREFERENCES

A. Device design

To design the device so the desired hydrodynamic resistances were achieved at each bifurcation, the device was modeled as an analogous resistive hydraulic circuit. The pressure–flow rate relationship at each node was defined by the hydraulic equivalent of Ohm's law,where ΔP is the pressure drop, R is the hydraulic resistance where both geometry and viscosity contribute, and Q is the flow rate. R was defined as flow in a rectangular duct as R=[4ab33μL(1−192bπ5a∑n=1,3,5,…∞tanh(nπa2b)n5)]−1,where μ is the fluid viscosity, L is the channel length, a is half the channel width, and b is half the channel depth.The device was designed as two sequential (upstream and downstream) bifurcations, which can be seen in Fig. 1(a). All channels had a depth of 6.5 μm. The feeder channel width was 8 μm wide and 250 μm in length. The feeder channel bifurcates into two upstream daughter channels, U1 and U2. Channel U1 was 6 μm wide with length varied between 50, 100 , and 300 μm long, while channel U2 had dimensions of 6 μm wide and was 250 or 350 μm long. Channel U1 further bifurcated into a second downstream bifurcation, channels D1 and D2 that were 6 μm wide with variable lengths to achieve the desired volumetric flow rate ratios, Q*s, were modulated by the daughter channels’ relative hydraulic resistances (see Sec. for a definition of how Q* is defined). Channels D1 and D2 both fed into the same outlet, while U2 drained to a separate outlet. Channel cross-sectional area dimensions were chosen to be compared with vessel diameters found in the microcirculation,11. S. S. Shevkoplyas, S. C. Gifford, T. Yoshida, and M. W. Bitensky, “Prototype of an in vitro model of the microcirculation,” Microvasc. Res. 65(2), 132–136 (2003). https://doi.org/10.1016/S0026-2862(02)00034-1 resulting in highly confined flow.The sequential bifurcation network was designed such that given the same pressure drop, ΔP, the downstream bifurcation would have a fixed volumetric flow rate ratio per device of either QDN* = 0.5, 0.66, or 0.8 (where N denotes the higher flow rate channel), while the upstream bifurcation would have the following volumetric flow rate ratios (QU1*'s): 0.2, 0.25, 0.33, 0.4, 0.5, 0.67, 0.75, 0.8. QU1* was controlled via the water column described in the experimental Sec. . QD* was controlled by modulating the relative resistance of channels D1 and D2 by changing the length of the channel. For example, a QD2* = 0.667 flow rate ratio was achieved by making the length of D2 = 449 and D1 = 224.5 μm, maintaining the equal resistance at the upstream bifurcation. QD2* = 0.8 was achieved by making the length of channel D2 = 750 and D1 = 188 μm. The channel lengths were chosen to maintain the same hydrodynamic resistance that occurred in the QD2* = 0.5 device.

Additional devices were designed with the length of U1 being either 50 or 300 μm. Maintaining equal hydrodynamic resistance at the upstream bifurcation necessitated modifying the lengths of channels U2 (LU2), D1 (LD1), and D2 (LD2). LU1 was reduced to 50 μm and the lengths of channels D1 and D2 were increased to 200 μm, thus maintaining QU* = 0.5 and the total device hydrodynamic resistance without necessitating a modification to LU2. Increasing LU1 to 300 μm required an increase of LU2 to 300 μm to maintain a 1:1 flow rate ratio at the upstream bifurcation. The lengths of both D1, and D2 were maintained at 50 μm long. This caused an increase in the overall hydraulic resistance, which necessitated an increase of ΔP to −8.17 cm of water to maintain an average feeder channel RBC velocity of 1.5 mm/s.

Network designs were verified via a COMSOL simulation of a Newtonian fluid flowing through each bifurcating network model as shown in Fig. 1(c).

B. Device fabrication

Microfluidic devices were fabricated using a standard soft lithography process.3333. D. C. Duffy, J. C. McDonald, O. J. A. Schueller, and G. M. Whitesides, “Rapid prototyping of microfluidic systems in poly(dimethylsiloxane),” Anal. Chem. 70(23), 4974–4984 (1998). https://doi.org/10.1021/ac980656z Devices were designed in AutoCAD and transferred to a chrome mask via direct write lithography (Front Range Photomask, Lake Havasu City, AZ). Relief features, 6.5 μm high, were defined on a silicon wafer using SU8-2007 photoresist (Kayaku Advanced Materials, Westborough, MA) through contact lithography (EVG620, EV Group, Tempe, AZ). Polydimethylsiloxane (PDMS, SYLGARD™ 184 Silicon Elastomer Kit, Dow Chemical Co., Midland, MI) was mixed at a ratio of 10:1 of elastomer to curing agent, degassed, poured over the wafer, and allowed to cure overnight in an oven at 65 °C. After curing, the PDMS devices were cut out using a razor blade and peeled from the relief mold. A 3 mm reservoir was punched at the channel inlet, while a 1.5 mm outlet was punched at each outlet. Devices were cleaned and bonded to a glass coverslip using oxygen plasma treatment. The surfaces of the PDMS microfluidic device and glass coverslips were treated under oxygen plasma at 100 W power, 250 SCCM O2, at 700 mTorr for 60 s (March PX-250 Plasma Treatment System, Nordson Corporation, Westlake, OH). Immediately following surface activation, PDMS was brought in contact with the glass coverslip and allowed to bond at 65 °C overnight.

C. Red blood cell sample preparation

Immediately prior to experiments, approximately 25 μl of whole human blood was collected in SAFE-T-FILL® Blood Collection Tubes (Ram Scientific, Nashville, TN) via lancet fingerstick from healthy, consenting volunteers in accordance with a protocol approved by the Rutgers Institutional Review Board for the protection of human subjects in research (eIRB Protocol No. Pro2020000433). RBC isolation was conducted as follows: whole blood is centrifuged at 1.8 g for 10 min, buffy coat aspirated, and washed in isotonic phosphate buffered solution (PBS). This process is repeated three times. RBCs are then resuspended in isotonic 1× PBS at either 2%, 20%, or 45% hematocrit. Experiments were run with 20% hematocrit unless stated otherwise. While the overall average hematocrit is 45%, in the circulatory system, the local hematocrit in the microcirculation is much lower, ranging from 0% to 36%.3434. H. H. Lipowsky, S. Usami, and S. Chien, “In vivo measurements of ‘apparent viscosity’ and microvessel hematocrit in the mesentery of the cat,” Microvasc. Res. 19(3), 297–319 (1980). https://doi.org/10.1016/0026-2862(80)90050-3 The suspension osmolality was confirmed to be within physiological ranges via an osmometer (model 3D3, Advanced Instruments, Norwood, MA).3535. S. N. Cheuvront, R. W. Kenefick, K. R. Heavens, and M. G. Spitz, “A comparison of whole blood and plasma osmolality and osmolarity,” J. Clin. Lab. Anal. 28(5), 368–373 (2014). https://doi.org/10.1002/jcla.21695 Hematocrit was confirmed via direct cell counting on a hemocytometer.

Once isolated, approximately 10%–15% of the RBCs were fluorescently stained to allow cell tracking by first resuspending the aliquot at a density of 106 cells/ml. Then, Vybrant® DiO (Invitrogen, Carlsbad, CA) was added at a concentration of 5 μl/ml. The solution was subsequently incubated at 45 °C for 2 h. Afterward, the stained RBCs were again centrifuged at 1.8 g for 5 min, the supernatant was aspirated, washed, and resuspended in 1× PBS. This process was repeated twice. Stained RBCs were then resuspended at the experimental hematocrit and added back to the sample.

D. Metrics

RBC partitioning was quantified as erythrocyte flux ratio (N*) at each bifurcation in the device and was analyzed as a function of the flow rate ratio of the daughter channel to the feeder channel (Q*). Q* and N* are defined as Q∗=∫d0⁡U¯⋅dA∫fs⁡U¯⋅dA,N∗=NdNf,where U̅ is the average channel velocity, A is the channel cross-sectional area, N is the number of RBCs that perfuse into the respective channel, subscript f refers to the feeder channel, and subscript d refers to daughter channel following the upstream or downstream bifurcation. All Q* and N* measurements at the upstream bifurcation were determined with respect to channel U1 (QU1*) and with respect to the higher flow rate channel at the downstream bifurcation. At the downstream bifurcation QD1* denotes daughter channel D1 received the higher flow rate and QD2* denotes channel D2 received the higher flow rate.

By this convention, at the upstream bifurcation a QU1* = 0.2 represents a 1:4 flow rate ratio between channel U1 and U2 with 20% of the volumetric flow rate proceeding into channel U1 toward the second bifurcation and 80% of the flow proceeding along channel U2 toward the outlet. A QU1* of 0.5 represents a 1:1 flow rate ratio with the flow equally split between channel U1 and channel U2. A QU1* of 0.8 is a 4:1 flow rate ratio between channel U1 and U2 with 80% of the volumetric flow rate proceeding into channel U1 and 20% of the flow proceeding along channel U2.

At the downstream bifurcation, QD* was calculated with respect to the higher flow rate daughter channel; a QD2* of 0.667 represents a 2:1 flow rate ratio between channel D2 and D1 with 66.7% of the volumetric flow rate proceeding into channel D2 [above the downstream bifurcation in Fig. 1(a)] and 33.3% of the flow proceeding along channel D1 [below the downstream bifurcation in Fig. 1(a)]. Similarly, a QD1* of 0.667 is the opposite at a 1:2 flow rate ratio between channel D2 and D1 with 33.3% of the blood flowing into channel D2 and 66.7% of the flow proceeding into channel D1.

Partitioning is quantified via N*. NU1* or ND2* of 0.5 shows the erythrocyte flux is equally balanced between the two daughter branches of either bifurcation whereas NU1* or ND2* > 0.5 implies a higher erythrocyte flux into channel U1 or D2, respectively, and NU1* or ND2* < 0.5 implies a higher erythrocyte flux into channel U2 or D1, respectively.

To investigate both the cause and effects of cell position on the erythrocyte flux ratio, N*, RBC/hematocrit skewness in the channel was parameterized via image processing. First, background subtraction was used to isolate RBCs within the images. The image was then thresholded and images were analyzed in the white box depicted in Fig. 1(a) and colored boxes in subsequent images of the devices for the presence of RBCs. Each individual pixel in the ROI was assigned a counter starting at 0. If an individual pixel was within the bounding area for an RBC in the ROI, the pixel counter was incremented by one for each frame that contained an RBC. This was done for the entire data set (t ∼ 10 s) within the white highlighted region denoted in Fig. 1(a). The combination of low channel height and single file flow in the z-direction enabled the use of this method.Once the pixel counts for the dataset were determined, the data were parameterized to RBC skewness (SH*)2323. J. M. Sherwood, D. Holmes, E. Kaliviotis, and S. Balabani, “Spatial distributions of red blood cells significantly alter local haemodynamics,” PLoS One 9(6), e100473 (2014). https://doi.org/10.1371/journal.pone.0100473 along a line within the ROI region, SH∗=∫w/2w⁡H(y∗)dy∗∫0w⁡H(y∗)dy∗−0.5,where w is the width of the channel and H(y∗) is the pixel counter data described above. Symmetric RBC position within the channel would result in an SH* = 0. A histogram is then plotted in the ROI.

When comparing ND* or SH* data at the downstream bifurcation with the NU* or SH* for comparable Q* at the upstream bifurcation (i.e., QU1* = 0.5 compared to QD2* = 0.5) an ANOVA was used for all statistical analyses with p values calculated for all comparisons. If the ANOVA resulted in statistical significance (p < 0.05), a Tukey honest significance test was then conducted to determine which data are statistically significant from each other. When only two conditions were being compared, a Student's t-test was used.

E. Experimental

Flow through the microfluidic network was controlled with a water column siphon-based pressure gradient as previously reported24,3024. A. Pskowski, P. Bagchi, and J. D. Zahn, “Investigation of red blood cell partitioning in an in vitro microvascular bifurcation,” Artif. Organs 45, 1083–1096 (2021). https://doi.org/10.1111/aor.1394130. J. M. Burns, X. Yang, O. Forouzan, J. M. Sosa, and S. S. Shevkoplyas, “Artificial microvascular network: A new tool for measuring rheologic properties of stored red blood cells,” Transfusion 52(5), 1010–1023 (2012). https://doi.org/10.1111/j.1537-2995.2011.03418.x Each outlet was connected to a separate 8.66 mm wide reservoir consisting of an open 3 ml syringe (Becton Dickinson, Franklin Lakes, NJ) connected to the device by 1.5 mm inner diameter tygon tubing (Cole-Parmer, Vernon Hills, IL), shown in Fig. 1(b). Adjustment of the height difference between the device entrance and each reservoir attached to the daughter channels at the outlet allowed an independent pressure drop and flow rate control of the flow in the upstream channels, U1 and U2. Average fluid velocity was held constant for all devices, regardless of channel size by modulating the driving pressure difference. Experiments were conducted with an average RBC velocity ranging from 1.3 to 2.7 mm/s, dependent upon reservoir hematocrit. Reservoir fluid was periodically aspirated with a P20 pipet to prevent RBC settling and prevent significant changes in local fluid viscosity due to changes in local RBC concentration.

Images of RBC perfusion through the microchannels were captured using an Olympus IX71 microscope with either a 20× or 40× objective and an attached CMOS PCO Edge 5.5 camera at frame rates of 350–420 fps. For cell tracking, epifluorescence microscopy with the FITC filter cube was used with the brightfield lamp slightly turned on. This enabled cell tracking of the stained RBCs as well visualization of the surrounding unstained RBCs. Images were later analyzed for cell velocity and number as well as the trajectory of stained cells using a custom MATLAB script. An average of 1117 RBCs were captured and analyzed for each individual experiment. DiO stained RBCsat high hematocrits were imaged via flourscence microscopy land ow light brightfield microscopy, enabling a sharp contrast between stained cells and the background while still visualizing the surrounding unstained RBCs.

Cell velocity was calculated as follows. The user selected a region of interest (ROI) in the desired channel (feeder, daughter 1) at t = t0. The ROI was thresholded and checked to see if an RBC was present. If no RBC was present the ROI was not used. A cross correlation was then performed using the ROI and an image t = t0 + 5 ms. This process was repeated for all images in the series and averaged to obtain the mean velocity.

We assume the experimental pressures and flow rates are linearly dependent on each other through the hydraulic resistance of the microchannels because the microchannels do not exhibit pressure/flow induced deformation or compliance at the applied pressures. Thus, the flow rate ratio (Q*) is modulated through pressure modulation and verified by calculating the flow rate in each of the daughter channels and Q* from the experimentally determined RBC velocities. For example, for a flow rate ratio, QU1*, of 0.667 in the upstream bifurcation at a flow rate of 0.0047 μl/min, the two water columns at the outlets were positioned at −7.07 and −5.14 cm to obtain siphon pressures of −693.33 and −504.06 Pa, respectively.

The maximum flow rate achieved using the syphon-based pressure modulation was 0.006 71 μl/min, which would cause an increase in the outlet reservoir height of 3.408 μm over an hour of constant perfusion, a negligible change in driving pressure. Reynolds number was based on the microchannel hydraulic diameter,Deq, (Deq=4AP, where A is the channel cross-sectional area and P is the wetted perimeter of the channel; 7.17 μm for the 8 × 6.5 μm2 channel device) and was calculated bywhere ρ is the solution density (1000 kg/m3 at room temperature), u is the characteristic flow velocity, and μ is the blood solution viscosity. Viscosity for a 20% hematocrit sample was estimated via the Einstein equationwhere μ0 is the viscosity of the 1× PBS suspending medium in the absence of RBCs (1 cP at 25 °C)3636. J. Rumble, in CRC Handbook of Chemistry and Physics, 100th Edition, 7th ed (CRC Press, 2019). and ϕ is the volume fraction (hematocrit) of RBCs (0.2). The maximum average RBC velocity ranged from 1.3 to 2.7 mm/s; prior work has shown that red blood cell velocities found in vivo are between 0.5 and1.6 mm/s in rat microcirculation;3737. D. Kleinfeld, P. P. Mitra, F. Helmchen, and W. Denk, “Fluctuations and stimulus-induced changes in blood flow observed in individual capillaries in layers 2 through 4 of rat neocortex,” Proc. Natl. Acad. Sci. U.S.A. 95(26), 15741–15746 (1998). https://doi.org/10.1073/pnas.95.26.15741 however, it can reach as high as 2–3 mm/s in brain capillaries.3838. E. Chaigneau, M. Roche, and S. Charpak, “Unbiased analysis method for measurement of Red blood cell size and velocity with laser scanning microscopy,” Front. Neurosci. 13, 644 (2019). https://doi.org/10.3389/fnins.2019.00644 This corresponds to a Re ranging from 0.0019 to 0.0079, showing dominance of viscous forces. Due to the Fahreus effect, the bulk fluid average velocity is less than the RBC velocity, confirming a dominance of viscous forces.8